Gravitational quantum switch on a superposition of spherical shells

In the absence of a complete theory of quantum gravity, phenomenological models built upon minimal assumptions have been explored for the analysis of possible quantum effects in gravitational systems. Implications of a superposition of geometries have been considered in such models, including the occurrence of processes with indefinite order. In a gravitational quantum switch, in particular, the order of operations applied by two agents on a target system is entangled with the state of the geometry. We consider a model describing the superposition of geometries produced by distinct arrangements of spherical mass shells, and show that a protocol for the implementation of a gravitational quantum switch can be formulated in such a system. The geometries in superposition are identical in an exterior region outside a given radius, and differ within such a radius. The exterior region provides a classical frame from which the superposition of geometries in the interior region can be probed. One of the agents crosses the interior region and becomes entangled with the geometry, which is explored as a resource for the implementation of the quantum switch. Novel features of the protocol include the superposition of nonisometric geometries, the existence of a region with a definite geometry, and the fact that the agent that experiences the superposition of geometries is in free fall, preventing information on the global geometry to be obtained from within its laboratory.